New Sharp Bounds for the Bernoulli Numbers and Refinement of Becker-Stark Inequalities

Abstract

We obtain new sharp bounds for the Bernoulli numbers: $2(2n)!/({\pi }^{2n}({2}^{2n}-1))<|{B}_{2n}|\le (2({2}^{2k}-1)/{2}^{2k})\zeta (2k)(2n)!/({\pi }^{2n}({2}^{2n}-1))$, $n=k,k+1,\dots , k\in {N}^{+}$, and establish sharpening of Papenfuss's inequalities, the refinements of Becker-Stark, andSteckin's inequalities. Finally, we show a new simple proof of Ruehr-Shafer inequality.